Core-collapse supernovae (CCSNe) are the luminous explosions that herald the death of massive stars. Neutron stars, pulsars, magnetars, and stellar-mass black holes are all born out of these explosions. Some Gamma-Ray Bursts (GRBs) have been associated with CCSNe, raising the possibility of a common progenitor for both. CCSNe are chiefly responsible for the production of elements heavier than iron throughout the universe; their importance in galactic chemical evolution cannot be underestimated. The first stars, expected to be relatively massive, likely ended as CCSNe as well. These bright events, occurring just a couple million years after the Big Bang, may be some of the most distant, observable objects in the universe with the upcoming James Webb Space Telescope. Despite the importance of CCSNe to our understanding of many aspects of the universe the mechanism that reverses stellar core collapse and drives supernova explosions is not fully understood. The CCSN mechanism is one of the most important challenges for modern computational astrophysics. I will discuss the current state-of-the-art of CCSN theory and simulation, with an emphasis on my recent work on three-dimensional CCSN simulations. I will highlight some of the most interesting and important questions supernova theorists are currently wrestling with, in particular the importance of fully three-dimensional simulations.